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A stochastic partial differential equation along the lines of the Kardar-Parisi-Zhang equation is introduced for the evolution of a growing interface in a radial geometry. Regular polygon solutions as well as radially symmetric solutions are identified in the deterministic limit. The polygon solutions, of relevance to on-lattice Eden growth from a seed in the zero-noise limit, are unstable in the continuum in favour of the symmetric solutions. The asymptotic surface width scaling for stochastic radial interface growth is investigated through numerical simulations and found to be characterized by the same scaling exponent as that for stochastic growth on a substrate.Comment: 12 pages, Elsevier style, 5 figure

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Detonations in non-ideal explosives tend to propagate significantly below the nominal one-dimensional detonation speed. In these cases, multi-dimensional effects within the reaction zone are important. A streamlinebased approach to steady-state non-ideal detonation theory is developed. It is shown in this study that, given the streamline shapes, the two-dimensional problem reduces to an ordinary differential equation eigenvalue problem along each streamline, the solution of which determines the local shock shape that, in turn, leads to the solution of the detonation speed as a function of charge diameter. A simple approximation of straight but diverging streamlines is considered. The results of the approximate theory are compared with those of high-resolution direct numerical simulations of the problem. It is shown that the straight streamline approximation is remarkably predictive of highly non-ideal explosive diameter effects. It is even predictive of failure diameters. Given this predictive capability, one potential use of the method is in the determination of rate law parameters by fitting to data from unconfined rate stick experiments. This is illustrated by using data for ammonium nitrate fuel oil explosives.

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In this paper, a one-dimensional stability analysis of weakly curved, quasi-steady detonation waves is performed using a numerical shooting method, for an idealized detonation with a single irreversible reaction. Neutral stability boundaries are determined and shown in an activation temperature-curvature diagram, and the dependence of the complex growth rates on curvature is investigated for several cases. It is shown that increasing curvature destabilizes detonation waves, and hence curved detonations can be unstable even when the planar front is stable. Even a small increase in curvature can significantly destabilize the wave. It is also shown that curved detonations are always unstable sufficiently near the critical curvature above which there are no underlying quasi-steady solutions.

The nonlinear stability of cylindrically and spherically expanding detonation waves is investigated using numerical simulations for both directly (blast) initiated detonations and cases where the simulations are initialized by placing quasi-steady solutions corresponding to different initial shock radii onto the grid. First, high-resolution onedimensional (axially or radially symmetric) simulations of pulsating detonations are performed. Emphasis is on comparing with the predictions of a recent one-dimensional linear stability analysis of weakly curved detonation waves. The simulations show that, in agreement with the linear analysis, increasing curvature has a rapid destabilizing effect on detonation waves. The initial size and growth rate of the pulsation amplitude decreases as the radius where the detonation first forms increases. The pulsations may reach a saturated nonlinear behaviour as the amplitude grows, such that the subsequent evolution is independent of the initial conditions. As the wave expands outwards towards higher (and hence more stable) radii, the nature of the saturated nonlinear dynamics evolves to that of more stable behaviour (e.g. the amplitude of the saturated nonlinear oscillation decreases, or for sufficiently unstable cases, the oscillations evolve from multi-mode to period-doubled to limit-cycle-type behaviour). For parameter regimes where the planar detonation is stable, the linear stability prediction of the neutrally stable curvature gives a good prediction of the location of the maximum amplitude (provided the stability boundary is reached before the oscillations saturate) and of the critical radius of formation above which no oscillations are seen. The linear analysis also predicts very well the dependence of the period on the radius, even in the saturated nonlinear regimes. Secondly, preliminary two-dimensional numerical simulations of expanding cellular detonations are performed, but it is shown that resolved and accurate calculations of the cellular dynamics are currently computationally prohibitive, even with a dynamically adaptive numerical scheme.